Hardy algebras associated with $W^*$-correspondences (point evaluation and Schur class functions)

Hardy Algebras, W*-Correspondences and Interpolation Theory

Given a von Neumann algebra $M$ and a $W^{\ast}$-correspondence $E$ over $M$, we construct an algebra $H^{\infty}(E)$ that we call the Hardy algebra of $E$. When $M=\mathbb{C}=E$, then $H^{\infty}(E)$ is the classical Hardy space $H^{\infty}(\mathbb{T})$ of bounded analytic functions on the unit disc. We show that given any faithful normal representation $σ$ of $M$ on a Hilbert space $H$ there is a natural correspondence $E^σ$ over the commutant $σ(M)^{\prime}$, called the $σ$-dual of $E$, and that $H^{\infty}(E)$ can be realized in terms of ($B(H)$-valued) functions on the open unit ball $\ma…

Generalized Inner-Outer Factorization in non commutative Hardy Algebras

Let $H^{\infty}(E)$ be a non commutative Hardy algebra, associated with a $W^*$-correspondence $E$. In this paper we construct factorizations of inner-outer type of the elements of $H^{\infty}(E)$ represented via the induced representation, and of the elements of its commutant. These factorizations generalize the classical inner-outer factorization of elements of $H^\infty(\mathbb{D})$. Our results also generalize some results that were obtained by several authors in some special cases.

Representations of Hardy Algebras: Absolute Continuity, Intertwiners and Superharmonic Operators

Suppose $\mathcal{T}_{+}(E)$ is the tensor algebra of a $W^{*}$-correspondence $E$ and $H^{\infty}(E)$ is the associated Hardy algebra. We investigate the problem of extending completely contractive representations of $\mathcal{T}_{+}(E)$ on a Hilbert space to ultra-weakly continuous completely contractive representations of $H^{\infty}(E)$ on the same Hilbert space. Our work extends the classical Sz.-Nagy - Foiaş functional calculus and more recent work by Davidson, Li and Pitts on the representation theory of Popescu's noncommutative disc algebra.

Progress in noncommutative function theory

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{*}$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the…

Test functions, Schur-Agler classes and transfer-function realizations: the matrix-valued setting

Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy alge…

Duality of $W^*$-correspondences and applications

This is an exposition of our work on the canonical models of representations of Hardy algebras associated with $W^*$-correspondences. It generalizes the model theory of Sz.-Nagy and Foias and the work of Popescu on models of row contractions. We also emphasize the concept of duality for $W^*$-correspondences and its relation to Morita equivalence of correspondenses.

矩阵代数上 $k$-(超)正映射的 Schoenberg 对应

Schoenberg Correspondence for $k$-(Super)Positive Maps on Matrix Algebras

本文证明非单位半群的 Schoenberg 型对应，刻画 $k$-正、$k$-超正及 $k$-纠缠破坏映射的生成元，并重证 LGKS 定理，通过实例展示正性随时间改善。